3. Consider a Markov chain on states {0, 1, 2, 3, 4, 5} with the transition probability matrix 0 1 2 3 4 5 0 1 00 1 0 2 3 4 000 38230 098910 ○ 147SOLCO 3418140 0 ○ ○ 1430 000 000 0 ст 5 00 001 Classify each state as transient or recurrent, and compute lim∞ P3 if it exists? 31
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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.Suppose the transition matrix for a Markov Chain is T = stable population, i.e. an x0₂ such that Tx = x. ساله داد Find a non-zero2. For a irreducible Markov chain with a stationary distribution 7, show that all the states are recurrent.
- Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).Construct a transition probability matrix of a Markov chain with state space {1, 2, . . . , 8} in which {1, 2, 3} is a transient class having period 3, {4} is an aperiodic transient class, and {5, 6, 7, 8} is a recurrent class having period 2.A Markov chain on states {1,2,3,4,5,6} has transition matrix 0 를 0 0 1 0 % 을 0 을 0 0 0 음 2 3 3 4 4 3 1 1 1 1 1 6 Find all communication classes and classify each class as either recurrent or transient.
- Suppose that a Markov chain with 4 states and with transition matrix P is in state 3 on the fifth observation. Which of the following expressions represents the probability that it will be in state 4 on the tenth observation? (A) the (5, 10) entry of Pª (B) the (3, 4) entry of Pš (C) the (10, 5) entry of Pª (D) the (4, 3) entry of P3 (E) the (5, 10) entry of P3 (F) the (4, 3) entry of P10 (G) the (3, 4) entry of P10 (H) the (10, 5) entry of P3Suppose that a Markov chain with 4 states and with transition matrix P is in state 4 on the fourth observation. Which of the following expressions represents the probability that it will be in state 1 on the sixth observation? (A) the (4, 6) entry of P (B) the (1,4) entry of P2 (C) the (4, 1) entry of P2 (D) the (4, 6) entry of P4 (E) the (1,4) entry of P6 (F) the (4, 1) entry of P6 (G) the (6,4) entry of P (H) the (6, 4) entry of P4 O A В C O D O E F O HSuppose that a Markov chain with 4 states and with transition matrix P is in state 3 on the fifth observation. Which of the following expressions represents the probability that it will be in state 4 on the tenth observation? (A) the (5, 10) entry of P4 (B) the (3, 4) entry of P (C) the (10, 5) entry of P4 (D) the (4, 3) entry of P5 (E) the (5, 10) entry of P3 (F) the (4,3) entry of P10 (G) the (3, 4) entry of P10 (H) the (10, 5) entry of P3 O A OB C O F OH
- 2.- Let X₁, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix 1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6, (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it. Your answer should refer to the relevant Theorems in the notes. (b) Find the limiting distribution for this Markov chain. (100) (c) Without doing any more calculations, what can you say about p₁,1 (100) ? and P2,1Can a Markov chain in general have an infinite number of states? O yes no Previous Sin51. Consider a Markov Chain with state space {0,1, 2, 3, 4} and transition matrix 2 3 4 1 1 1/3 1/3 1/3 .0 1/3 1/3 1/3 0 1/3 1/3 1/3 0 - 0 0 1 P = 2 3 (n) Then limn-00 P23 equals 1 1. 3 2. 3: 0 4. 1 4-
